Kinetics
5.2.1 Force, Traction and Stress Vectors 38 There are two kinds of forces in continuum mechanics body forces act on the elements of volume or mass inside the body, e.g. gravity, electromagnetic fields. dF pbdVol. surface forces are contact forces acting on the free body at its bounding surface. Those will be defined in terms of force per unit area. 39 The surface force per unit area acting on an element dS is called traction or more accurately stress vector. f tdS if txdS jf tydS kf tzdS 5.37...
Fundamental Laws of Continuum Mechanics
58 We have thus far studied the stress tensors Cauchy, Piola Kirchoff , and several other tensors which describe strain at a point. In general, those tensors will vary from point to point and represent a tensor field. 59 We have also obtained only one differential equation, that was the compatibility equation. 60 In this chapter, we will derive additional differential equations governing the way stress and deformation vary at a point and with time. They will apply to any continuous medium, and...
Review of Continuum Mechanics Stress
2.2.1.1 Hydrostatic and Deviatoric Stress Tensors If we let a denote the mean normal stress p o -p i lt 7n 722 f 33 U tr a 2.17 then the stress tensor can be written as the sum of two tensors Hydrostatic stress in which each normal stress is equal to p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. _ Deviatoric Stress which causes the change in shape. 2.2.1.2 Geometric Representation of Stress States 23 Using the...
Elliptical hole in a Uniformly Stressed Plate Inglis
24 Next we consider the problem of an elliptical hole in an infinite plate under uniform stress, Fig. 6.2. Adopting the curvilinear coordinate system described in sect. 5.8, we define a and b as the major and minor semi-axes respectively. The elliptical hole is itself defined along a ao, and as we go around the ellipse varies from 0 to 2n. 25 Thus substituting fJ 0 and 3 in Eq. 5.139 and 5.140 we obtain x p o a c cosh a cos 3 c cosh a0 a c cosh a0 x2 p n 2 b c sinh a sin 3 c sinh a0 b c sinh a0...
a Vmd
Figure 2.1 Typical Stress-Strain Curve of an Elastoplastic Bar 8 Up to the A0, the response is linearly elastic, and unloading follows the initial loading path. O A represents the elastic range where the behavior is load path independent. 9 At point Ao, the material has reached its elastic limit, from Ao to C the material becomes plastic and behaves irreversibly. In this plastic range, the stiffness decreases progressively, and eventually fails at C. 10 Unloading from any point between A0 and C...
Stress Intensity Factors
12 As shown in the preceding chapter, analytic derivation of the stress intensity factors of even the simplest problem can be quite challenging. This explain the interest developed by some mathematician in solving fracture related problems. Fortunately, a number of simple problems have been solved and their analytic solution is found in stress intensity factor handbooks. The most commonly referenced ones are Tada, Paris and Irwin's Tada et al. 1973 , and Roorke and Cartwright, Rooke and...
s
Figure 2.8 Isotropic Hardening Softening Fn 0 for Hp 0 Perfectly Plastic Figure 2.8 Isotropic Hardening Softening 2.5.1 Isotropic Hardening Softening J2 plasticity 52 In isotropic hardening softening the yield surface may shrink softening or expand hardening uniformly see figure 2.8 . 1. Yield function for linear strain hardening softening dF dF dq F s -4 A 0 from which we solve the plastic multiplier 3. Tangential stress-strain relation deviatoric 53 Note that isotropic hardening softening is...
List of Figures
1.1 Kinematics of Continuous and Discontinuous Failure Processes 1 1.2 Discrete-Smeared Crack 2.1 Typical Stress-Strain Curve of an Elastoplastic 2.2 Elastoplastic Rheological Model for Overstress 2.3 Bauschinger Effect on Reversed 2.4 Stress and Strain Increments in Elasto-Plastic 2.5 Stress-Strain diagram for Elastoplasticity 2.6 Haigh-Westergaard Stress Space 2.7 General Yield 2.8 Isotropic 2.9 Kinematic 4.1 Cracked Cantilevered 4.2 Failure Envelope for a Cracked Cantilevered 4.3 Generalized...
J Plasticityvon Mises Plasticity
51 For J2 plasticity or von Mises plasticity, our stress function is perfectly plastic. Recall perfectly plastic materials have a total modulus of elasticity Et which is equivalent to zero. We will deal now with deviatoric stress and strain for the J2 plasticity stress function. since lt 7 0 in perfect plasticity, the second term drops out and F becomes S 2G e 2G - p 2.71 F 2Gs - m 0 2.73 4. Tangential stress-strain relation deviatoric Now we have the simplified expression Gep 2G 4 2-78 is the...
Griffith Theory
17 Around 1920, Griffith was exploring the theoretical strength of solids by performing a series of experiments on glass rods of various diameters. He observed that the tensile strength at of glass decreased with an increase in diameter, and that for a diameter 0 t in., at 500,000 psi furthermore, by extrapolation to zero diameter he obtained a theoretical maximum strength of approximately 1,600,000 psi, and on the other hand for very large diameters the asymptotic values was around 25,000 psi....
Circular Hole Kirsch
8 Analysing the infinite plate under uniform tension with a circular hole of diameter a, and subjected to a uniform stress a0, Fig. 6.1. 9 The peculiarity of this problem is that the far-field boundary conditions are better expressed in cartesian coordinates, whereas the ones around the hole should be written in polar coordinate system. 10 We will solve this problem by replacing the plate with a thick tube subjected to two different set of loads. The first one is a thick cylinder subjected to...
Crack Tip Opening Displacements
7 Within the assumptions and limitations of LEFM we have two valid and equivalent criteria for crack propagation 1 k vs kic which is a local criteria based on the strength of the stress singularity at the tip of the crack and 2 g vs gic or R which is a global criteria based on the amount of energy released or consumed during a unit surface crack's propagations. 8 In many cases it is found that LEFM based criteria is either too conservative and expensive as it does not account for plastification...
Crack Westergaard
35 Just as both Kolosoff 1910 and Inglis 1913 independently solved the problem of an elliptical hole, there are two classical solutions for the crack problem. The first one was proposed by Westergaard, and the later by Williams. Whereas the first one is simpler to follow, the second has the advantage of being extended to cracks at the interface of two different homogeneous isotropic materials and be applicable for V notches. 36 Let us consider an infinite plate subjected to uniform biaxial...
Tensors
10 We now seek to generalize the concept of a vector by introducing the tensor T , which essentially exists to operate on vectors v to produce other vectors or on tensors to produce other tensors . We designate this operation by T-v or simply Tv. 11 We hereby adopt the dyadic notation for tensors as linear vector operators 12 Whereas a tensor is essentially an operator on vectors or other tensors , it is also a physical quantity, independent of any particular coordinate system yet specified...
Lefm Design Examples
7 Following the detailed coverage of the derivation of the linear elastic stress field around a crack tip, and the introduction of the concept of a stress intensity factor in the preceding chapter, we now seek to apply those equations to some pure mode I practical design problems. 8 First we shall examine how is linear elastic fracture mechanics LEFM effectively used in design examples, then we shall give analytical solutions to some simple commonly used test geometries, followed by a...
ENERGY TRANSFER in CRACK GROWTH Griffith II
7 In the preceding chapters, we have focused on the singular stress field around a crack tip. On this basis, a criteria for crack propagation, based on the strength of the singularity was first developed and then used in practical problems. 8 An alternative to this approach, is one based on energy transfer or release , which occurs during crack propagation. This dual approach will be developed in this chapter. 9 Griffith's main achievement, in providing a basis for the fracture strengths of...
Energy Release Rate Equivalence with Stress Intensity Factor
42 We showed in the previous section that a transfer of energy has to occur for crack propagation. Energy is needed to create new surfaces, and this energy is provided by either release of strain 2 More rigorous estimates will be made in Chapter 11. energy only, or a combination of strain energy and external work. It remains to quantify energy in terms of the stress intensity factors. 43 In his original derivation, Eq. 9.25, Griffith used Inglis solution to determine the energy released. His...
J Integral
7 Eshelby Eshelby 1974 has defined a number of contour integrals that are path independent by virtue of the theorem of energy conservation. The two-dimensional form of one of these integrals can be written as where w is the strain energy density r is a closed contour followed counter-clockwise, as shown in Fig. 13.1 t is the traction vector on a plane defined by the outward drawn normal n and t an u the displacement vector, and dr is the element of the arc along the path r. Figure 13.1 j...
Plane Strain vs Plane Stress
36 Irrespective of a plate thickness, there is a gradual decrease in size of the plastic zone from the plate surface plane stress to the interior plane strain , Fig. 11.8. 37 The ratio of the plastic zone size to the plate thickness must be much smaller than unity for plane strain to prevail. It has been experimentally shown that this ratio should be less than 0.025. _ Figure 11.8 Plastic Zone Size Across Plate Thickness 38 We also observe that since rp is proportional to the plate thickness...